Results 1 to 2 of 2

Math Help - Prove the Bonferroni inequality

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    24

    Prove the Bonferroni inequality

    Hi,

    I would like to know how to prove the Bonferroni inequatlity (I have actually started but at some point, I don't know how to prove).

    1- P(A1) - P(A2)- P(A3)

    Note:

    Proof:


    = 1 - P(A1 U A2 U A3)
    = 1 - P(A1) - P(A2) - P(A3) + P(A1 n A2)
    + P(A1 n A3) + P(A2 n A3)
    - P(A1 n A2 n A3) (1)
    1- P(A1) - P(A2)- P(A3) (2)

    I would like to know how to justify, proof that
    P(A1 n A2) + P(A1 n A3) + P(A2 n A3) - P(A1 n A2 n A3) is positive?
    Like how do I go from the step before last step to my last step, that is from (1) to (2)?
    Thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    This follows from

    P(\cup A_n)\le \sum P(A_n)

    If you wish to see why 'P(A1 n A2) + P(A1 n A3) + P(A2 n A3) - P(A1 n A2 n A3) is positive'
    draw a Venn diagram http://www.learnnc.org/reference/Venn%20diagram
    The triple intersection is contained repeatedly in those other probabilities.

    Note that if A\subset B then P(A)\le P(B).

    This comes from P(B)=P(AB)+P(A'B)\ge P(AB)=P(A).
    Last edited by matheagle; August 3rd 2009 at 11:10 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Bonferroni's Inequality Proof
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: September 19th 2010, 08:00 AM
  2. Bonferroni Inequalities
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: September 18th 2010, 03:45 AM
  3. Bonferroni's inequality
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 14th 2010, 09:35 AM
  4. simplified prove of bonferroni inequality
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: January 29th 2010, 01:01 PM
  5. Proving Bonferroni's Inequality
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: February 28th 2008, 08:42 AM

Search Tags


/mathhelpforum @mathhelpforum